Sharp discrete inequalities and applications to discrete variational problems

The discrete Euler equation is established as a necessary condition for the solution of discrete variational problems for both infinite and finite time sequences. The variational criterion (infinite sum) is treated as a functional on l 1 n, the normed linear apace of infinite sequences. The discrete Euler equation is also found for the constrained discrete variational problem, i.e. for problems in which the extremal sequences must satisfy an additional subsidiary condition. Such a condition is assumed to be of a form that is readily applicable to discrete control problems, since it includes difference equations as a special case. The discrete variational problem with finite time (finite sum criterion) is treated as a special case of the infinite time.

This paper proposes a Sequential Approximate Optimization (SAO) for the discrete design variable problems using Radial Basis Function (RBF) Network. We consider that there are two important keys for SAO: One is the parameter adjustment for good approximation, and the other is to explore the sparse region for global approximation. The authors have proposed the simple estimate of the width in the Gaussian Kernel for good approximation. In addition, in order to explore the sparse region, we have developed the density function. The density function with the simple estimate of the width works well in the case of the continuous design variables. However, a simple application of the density function to the discrete design variable problems will cause some difficulties. In order to overcome these difficulties and find the sparse region of the discrete design variables with the density function, the new variables for the discrete design variables are introduced. By using the new variables, it is possible to find the sparse region of the discrete design variables. A simple sampling algorithm is shown, in which the Discrete Differential Evolution (DDE) for the discrete design variables is employed. Through typical numerical examples, the validity of proposed approach is examined.

At least since 1904 we know about the existence of singular monotone functions, i.e. continuous monotone functions of one real variable with vanishing derivative a.e. In that year Lebesgue [7] and Minkowski [8] gave their famous examples. In 1910 Faber [4], in 1916 Sierpinski [10] and then many others greatly enriched the variety of known examples. By integrating anyone of these functions we get a differentiable convex function with vanishing second derivative a.e. Thus we obtain smooth convex curves with vanishing curvature a.e. A very beautiful and simple example is due to de R h a m [3J: Take a convex polygon, consider on each side the two points dividing it into three equal parts and take the convex polygon with all these division points as vertices. Repeat the procedure. As a limit we get a smooth, strictly convex curve with a.e. vanishing curvature! The purpose of this paper is to show that, in the sense of Baire categories, most convex curves have the above property. More generally, most convex surfaces in IR" have a.e. a vanishing sectional curvature in every tangent direction. We start with some definitions. By a convex surface we shall always understand a closed convex surface in Busemann's sense (see [11, p. 3). A convex curve is a one-dimensional convex surface. A point x of a convex surface S is called smooth if S is differentiable at x. S is smooth if each of its points is smooth. Any half-line in a supporting plane (this is a hyperplane, see [1], p. 4) of a convex surface S, originating at a point x ~ S will be called supporting direction at x. A supporting direction at a smooth point is called tangent direction. The union of a circular with a square 2-cell, such that the radius and centre of the circle coincide with the side-length and a vertex of the square, is called corner-disk. The two points lying on both the circle and the square are called touching points of the corner-disk. A 2-cell having the union of half a circle with a segment as boundary is called semidisk. The points of the boundary of a semidisk which are not smooth are called corners of the semidisk. The radius of a corner-disk or a semidisk is the radius of the circle appearing in their definitions.

Discrete isoperimetric variational problems which model single and double integral isoperimetric problems are formulated and some multiplier rules are derived. For quardratic functionals, the Euler-Lagrange equation is linear and can be analyzed bydeterminant methods. difference equations methods. or numerically bythealgebraic eigenvalue problem. A specific example is given wherethe eigenfuntions and eigenvalue of this discrete problem converge. as the step size goes to zero, to the eigenfunctionsand eigenvalues of the corresponding, continuous problem Recent developments in algebraic geometry offer the hopeof using Groebner (=Gröbner or Grobner) basis methods to numerically solve some systems of equationsgenerated by Lagrange multiplier methods. These methods mayapply to nonlinear systems of equations whenever they can be reduced to systems of polynomial equations. The Groebner basis algorithm of Buchberger is an extension of Gaussian elimination to polynomial systems.

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

Symmetric duality for left and right Riemann–Liouville and Caputo fractional differences

For calculating constrained optimization problem various socio/bio-inspired algorithms have adopted a penalty function approach to handle linear and nonlinear constraints. In a general sense, the approach is quite easy to understand, but a precise choice of penalty parameter is very much important. It requires a bunch number of primer preliminaries. So as to beat this restriction another self-adaptive penalty function (SAPF) approach will be proposed and incorporated into Particle Swarm Optimization (PSO) algorithm. This approach is referred to as PSO-SAPF. Besides, PSO-SAPF approach will be hybridized with Colliding Bodies Optimization (CBO) referred to as PSO-SAPF-CBO algorithm. The performance of PSO-SAPF and PSO-SAPF-CBO algorithm will be distinctly validated by solving discrete and mixed variable problems from truss structure domain and linear and nonlinear domain. The effect of behavior of penalty parameter, penalty function and constrained violation will be analyzed and discussed with the advantages over other algorithms.

Quantum computing has become a pivotal innovation in computational science, offering novel avenues for tackling the increasingly complex and high-dimensional optimization challenges inherent in engineering design. This paradigm shift is particularly pertinent in the domain of structural optimization, where the intricate interplay of design variables and constraints necessitates advanced computational strategies. In this vein, the gate-based variational quantum algorithm utilizes quantum superposition and entanglement to improve search efficiency in large solution spaces. This paper delves into the gate-based variational quantum algorithm for the discrete variable truss structure size optimization problem. By reformulating this optimization challenge into a quadratic, unconstrained binary optimization framework, we bridge the gap between the discrete nature of engineering optimization tasks and the quantum computational paradigm. A detailed algorithm is outlined, encompassing the translation of the truss optimization problem into the quantum problem, the initialization and iterative evolution of a quantum circuit tailored to this problem, and the integration of classical optimization techniques for parameter tuning. The proposed approach demonstrates the feasibility and potential of quantum computing to transform engineering design and optimization, with numerical experiments validating the effectiveness of the method and paving the way for future explorations in quantum-assisted engineering optimizations.

Optimal damping of plates (or beams) partially covered by viscoelastic constrained layers is presented. The design variables are the locations and the sizes of the damped parts. The objective function is a linear combination of the first modal damping factors calculated from a specific finite element analysis. The discrete design variable optimization problem is solved using a genetic algorithm.

Topology optimization of pretensioned reflector antennas with unified cable-bar model

An approach to the sensitivity analysis for discrete optimization problems with perturbedobjective function is presented. The problem is stated in the following general form:minΣe∈Yc(e) : Y ∈ F, where c = (c(e), e ∈ E) is a vector of weights for some finite set E,and F ⫅ 2E is a given family of feasible subsets. It is assumed that the set of feasiblesolutions F is fixed, but the coefficients of the vector c may vary. The main problem consideredin this paper concerns the following question: How does the maximum relative errorof a given feasible solution depend on inaccuracies of the problem data? Two particularperturbations of the vector c are considered:(i) it is assumed that c is in a closed ball with radius δ≥0 and center co≥0 or(ii) the relative deviation of c(e) from the value co(e) is not greater than δ for any e ∈ E, i.e.,|c(e) ‐ co(e)| ≤δ co(e), e ∈ E.For a given feasible solution X ∈ F , two functions of the parameter δ are introduced: thesensitivity function s(X, δ) and the accuracy function a(X, δ). The values of s(X, δ) anda(X, δ) are equal to the maximum relative error of the solution X when the perturbations ofproblem data are of the type (i) or (ii), respectively. Some general, but computationallyinefficient formulae for calculating these functions are derived and their properties arestudied. It is shown that s(X, δ) and a(X, δ) are nondecreasing, convex functions with alimited number of breakpoints. A practically efficient method of calculating upper and lowerenvelopes for the accuracy function is presented. This method is based on the notion of k‐bestsolutions of the problem. It gives an interval to which the maximum relative error ofthe solution X must belong when the coefficients of vector c are given with accuracy δ. Theapproach is illustrated with an example of the symmetric traveling salesman problem.

The purpose of the current paper is to investigate several various problems for the categories STLs,SNe∗, and other related categories such as various new outcomes for the coefficients of f, together with majorization issues, the Hankel determinant, and the logarithmic coefficients with sharp inequalities and differential subordination implications.

A convex function f : [a,b] → ℝ satisfies the so-called Hermite-Hadamard inequality f (a + b 2 ) ≤ 1 b − a∫ abf(t)dt ≤f(a) + f(b) 2 . Motivated by the above estimates, in this paper we consider approximately monotone and convex functions, and give upper and lower bounds to the numerical integral mean, i.e., to 1 b−aIn(f), where In(f) denotes some of the most popular Newton-Cotes quadrature formulas.

This article explores the algorithms for solving individual classes of discrete extremal problems to find the exact optimum. When obtaining algorithms, the procedures for decoding and finding the maximum upper zero of discrete monotonic functions are used. Formulations of the tasks of decoding and searching for the maximum upper zero of discrete monotonic functions are formulated. To solve the problems of decoding and searching for the maximum upper zero of individual classes of 3-digit monotonic functions, estimates are given for the complexity of deciphering monotonic functions in a class.

We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations.